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Particle number fluctuations

In the grand canonical ensemble, the particle number N is not constant. It is, therefore, instructive to calculate the fluctuation in this quantity. As usual, this is defined to be

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Note that

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Thus,

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In order to calculate this derivative, it is useful to introduce the Helmholtz free energy per particle defined as follows:

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where tex2html_wrap_inline621 is the volume per particle.

The chemical potential is defined by

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Similarly, the pressure is given by

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Also,

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Therefore,

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and

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But recall the definition of the isothermal compressibility:

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Thus,

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and

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and the relative fluctuation is given by

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Therefore, in the thermodynamic limit, the particle number fluctuations vanish, and the grand canonical ensemble is equivalent to the canonical ensemble.



Mark Tuckerman
Tue Feb 1 14:50:00 EST 2000